Strange Attractors: General 2D Map - Part 1

by Antonio Sánchez Chinchón

An R experiment to create images generated by the trajectory of a particle according to a strange attractor.

Made with Rcpp, tidyverse

Blog post explaining the experiment: Rcpp, Camarón de la Isla and the Beauty of Maths

Inspired by: Strange Attractors: Creating Patterns in Chaos, by Julien C. Sprott

Github repo with more details

library(Rcpp)
library(tidyverse)

opt <-  theme(legend.position  = "none",
              panel.background = element_rect(fill="white", color="black"),
              plot.background  = element_rect(fill="white"),
              axis.ticks       = element_blank(),
              panel.grid       = element_blank(),
              axis.title       = element_blank(),
              axis.text        = element_blank())

cppFunction('DataFrame createTrajectory(int n, double x0, double y0, 
            double a1, double a2, double a3, double a4, double a5, 
            double a6, double a7, double a8, double a9, double a10, 
            double a11, double a12, double a13, double a14) {
            // create the columns
            NumericVector x(n);
            NumericVector y(n);
            x[0]=x0;
            y[0]=y0;
            for(int i = 1; i < n; ++i) {
            x[i] = a1+a2*x[i-1]+ a3*y[i-1]+ a4*pow(fabs(x[i-1]), a5)+ a6*pow(fabs(y[i-1]), a7);
            y[i] = a8+a9*x[i-1]+ a10*y[i-1]+ a11*pow(fabs(x[i-1]), a12)+ a13*pow(fabs(y[i-1]), a14);
            }
            // return a new data frame
            return DataFrame::create(_["x"]= x, _["y"]= y);
            }
            ')
a1 <- -0.6446
a2 <- -0.6085
a3 <- 0.6684
a4 <- -1.0413
a5 <- 0.5945
a6 <- 1.0694
a7 <- -0.9064
a8 <- -1.0151
a9 <- -0.6356
a10 <- -0.64460
a11 <- -0.64461
a12 <- -0.64462
a13 <- -0.64463
a14 <- -0.64464

df <- createTrajectory(10000000, 1, 1, a1, a2, a3, a4, a5, a6, 
                       a7, a8, a9, a10, a11, a12, a13, a14)

mx <- quantile(df$x, probs = 0.05)
Mx <- quantile(df$x, probs = 0.95)
my <- quantile(df$y, probs = 0.05)
My <- quantile(df$y, probs = 0.95)

df %>% filter(x > mx, x < Mx, y > my, y < My) -> df

plot <- ggplot(df) +
  geom_point(aes(x, y), shape=46, alpha=0.01, size=0, color="black") +
  scale_x_continuous(expand = c(0,0))+
  scale_y_continuous(expand = c(0,0))+
  coord_fixed() + 
  opt

plot


Compiled: 2019-04-18